I like
Nora well enough, though I prefer
Eleanor (for the -nor) and
Laura (for the sound). So I'd never use it. I can't see the point of sticking an -h on, but then I've seen more nightmareish ones like Diannah, Victoriah and no doubt others that I've forgotten. What makes those, and
Norah, bad is that the -h makes no difference to the pronunciation so it's purely ornamental or intended to be. And that doesn't work for me at all. At least
Mariah waves a flag and says "Pronounce me the old-fashioned way, because I'm so trendy!"